When \(m=1\) , Glendinning and Sidorov [1]} showed that \(\mathcal {U}_q\)  is countable for \(q\in (\mathcal {G},q_{KL})\)  and uncountable with positive Hausdorff
dimension for \(q\in (q_{KL},2]\) , where \(\mathcal {G}=(1+\sqrt{5})/2\)  is the golden radio, \(q_{KL}\)  know as the Komornik-Loreti constant (see [2]}).
Komornik and Loreti [3]}, de Vries, Komornik and Loreti [4]} proved that \(\mathcal {U}\)  is closed from above but not from below, and its closure \(\overline{\mathcal {U}}\)  is a Cantor set. Komornik and Loreti [2]}, [6]} proved that \(\mathcal {U}\)  has a smallest element \(q_{KL}\)  known as the Komornik-Loreti constant. Kong and Li [7]}, Komornik, Kong and Li [8]} determined the Hausdorff dimensions of \(\mathcal {U}_q\) .
Furthermore \((1,m+1]\setminus \overline{\mathcal {U}}=\cup (p_0,p_0^*)\) , where \(p_0\)  runs over \(\left\lbrace 1\right\rbrace \cup (\overline{\mathcal {U}}\setminus \mathcal {U})\)  and \(p_0^*\)  runs over a proper subset of \(\overline{\mathcal {U}}\) , and \(\mathcal {V}\cap (p_0,p_0^*)=\left\lbrace q_\ell :\ell =1,2\cdots \right\rbrace \)  is a strictly increasing sequence converging to \(p_0^*\) . Especially, \(\mathcal {V}\cap (1,q_{KL})\)  has a smallest element \(\mathcal {G}(m)\)  known as the generalized golden ratio, which was given by Baker [9]}. More exactly,
\(\begin{split}\mathcal {G}(m)=\left\lbrace \begin{array}{ll}k+1, & \text{if}\; m=2k, k=1,2,\cdots \\\frac{k+1+\sqrt{k^{2}+6k+5}}{2}, & \text{if}\; m=2k+1, k=0,1,\cdots \\\end{array}\right.\end{split}\) 
